In another direction it could be interesting to see if
our results might be used to help define
and study combinatorial landscapes of \(k\) -noncrossing RNA
structures. Combinatorial landscape theory involves the study of
configuration spaces (e.g. the space of RNA sequences with fixed length),
that come equipped with some notion
of adjacency (e.g. RNA sequences differing in one nucleotide are adjacent),
on which some fitness map is defined that
assigns a real number to each element in the space
(e.g. the energy of a minimum free energy structure for an RNA sequence) [1]}.
This concept is useful in, for example, combinatorial optimization
where it can be used to design algorithms for finding (locally) optimal elements
in the underlying space. There is a natural notion of adjacency for multitriangulations that is
given in terms of “flipping" around an edge in a multitriangulation to give another multitriangulation
(this generalizes the well-known notion of
flippings in triangulations – see e.g. [2]}).
Moreover,
any two multitriangulations in \(\mathcal {T}_{m,k}\)  can be converted from one to the other through a sequence of flips (see e.g. [3]} and the references therein).
Thus, it would be interesting to explore
if these results coupled with our above observations concerning multitriangulations
might be used as a starting point for investigating
landscapes of \(k\) -noncrossing RNA structures.
